Topological Devil's staircase in a constrained kagome Ising antiferromagnet

TL;DR

This work analyzes a kagome lattice Ising antiferromagnet under constrained, infinite first- and third-neighbor couplings and reveals an infinite sequence of thermal first-order transitions driven by condensation of system-spanning defect lines, analogous to a Kasteleyn transition but with a topological staircase. By mapping ground states to A-, B-, and C- strings (with DDWs) and exploiting a tensor-network CTMRG framework, the authors demonstrate that the ratio $n_C/n_A$ between defect types exhibits integer plateaus that define a devil's staircase not governed by commensurate wave-vectors. Real-space correlations, magnetic structure factors, and symmetry-order parameters are computed, revealing highly anisotropic, algebraically decaying correlations and a cascade of topological transitions as temperature increases. The results highlight a topological mechanism for a devil's staircase in constrained classical spin systems and point to potential extensions to finite $J_1,J_3$ and experimental realizations in magnetic oxides, artificial spin systems, or Rydberg-atom arrays.

Abstract

We show that the constrained Ising model on the kagome lattice with infinite first and third neighbor couplings undergoes an infinite series of thermal first-order transitions at which, as in the Kasteleyn transition, linear defects of infinite length condense. However, their density undergoes abrupt jumps because of the peculiar structure of the low temperature phase, which is only partially ordered and hosts a finite density of zero-energy domain walls. The number of linear defects between consecutive zero-energy domain walls is quantized to integer values, leading to a devil's staircase of topological origin. By contrast to the devil's staircase of the ANNNI and related models, the wave-vector is not fixed to commensurate values inside each phase.

Figures (16)

• Figure 1: Spin configurations of the kagome lattice Ising antiferromagnet Eq. \ref{['eq:ham']} are represented by blue and red dots. Orange segments show an alternative representation. (a) Main family of strings ground states : domains of left- and right-pointing orange arrows are delimited by A and B strings. The first dense and sparse rows which make up the Kagome lattice are highlighted as solid or dashed gray lines. Inset: labeling of the isotropic Ising couplings. (b) Zero-energy double domain walls (DDWs) correspond to arrows rotated all in the same direction and are entropically suppressed in the thermodynamic limit. A thick purple line highlights the center of each DDW. (c) C-string excitations in the first plateau are characterized by different orientations of the arrows inside the DDWs, which introduce an energy cost. The density of C-strings ($n_C$) is equal to the density of ferromagnetic nearest-neighbors on a horizontal dense row.
• Figure 2: Numerical results from CTMRG for $\chi = 100$ suggest a topological devil's staircase. Only temperatures where the results are converged in bond dimension are shown (for more details on the bond-dimension dependence of CTMRG results, see sm). (a) Energy. (b) Ratio of the strings density, revealing well-defined, integer-valued plateaus. (c) Density of A and C strings, related to the $\mathbb{Z}_3$ order parameter (see \ref{['sec:oderparameter']}). A careful inspection reveals a continuous temperature dependence in the apparent plateaus of $n_C$, as shown in the two insets.
• Figure 3: Topological plateaus are shown for increasing temperatures.(a-c): snapshots sampled from the CTM environment ($\chi=80)$ for (a) $p = n_C/n_A = 1$ ($T=3 J_2$) (b) $p= 2$, $T=3.4 J_2$ (c) $p = 3$, $T=3.8 J_2$ plateaus, with C-lines and green decorations. The number $p$ of C-lines between A strings is easily evaluated as the number of vertical orange bonds crossing a dense row of spins. Second row (d-f): corresponding magnetic structure factors exhibiting $n_C/n_A+2$ peaks at distances $2\pi/\bar{l}_1$. The structure factors are obtained by averaging over more than 2700 snapshots of size $40\times 40$ and clearly highlight the anisotropy of correlations, also evident in the real-space plots sm.
• Figure EM1: Local configurations of the kagome Ising antiferromagnet in the constrained limit of infinite $J_1$ and $J_3$. Top row: spins on white sites can be flipped with no cost, in which case the two possible alternatives for the dual bonds are dashed. The honeycomb lattice six-state clock-model mapping (black arrows) is defined by attributing to each triangle the direction of the arrow touching it. For the star (rightmost panel) there would be two possible choices; we systematically chose this one by convention. Bottom row: examples of mapping to DWs. Close to hexagons with K's or stars, different spin configurations map onto the same string configuration, contributing to the entropic gain of C lines.
• Figure EM2: Tensor network. (a) CTRMG idea: the environment surrounding a cluster is approximated by corner ($C_i$) and edge ($E_i$) tensors, which are truncated to a cutoff dimension $\chi$. We use single-site directional CTMRG orus_simulation_2009 with the isometries introduced in Ref. corboz_2014_isometries. (b) Each tensor in the network contains the Boltzmann weight of a kagome star vanhecke_solving_2021. Contraction of the legs of the tensors impose the matching of spin configurations on overlapping sites. (c) If configurations related by a global spin-flip symmetry are identified, there are 4 different states for each triangle. We represent them by drawing a dual lattice bond between adjacent spins pointing in the same direction. The last one is not allowed in the constrained limit ($J_1,J_3\rightarrow\infty$).